Given that `int_(0)^(pi//2) sin^(4) x cos^(2) x dx= (pi)/(32)`, then `int_(0)^(pi//2) cos^(4) x sin^(2) x dx`=

Options

A. `(pi)/(32)`
B. `(3pi)/(32)`
C. `(pi)/(2)`
D. none

Answer

To solve the problem, we need to find the value of the integral

\[ I = \int_{0}^{\frac{\pi}{2}} \cos^4 x \sin^2 x \, dx \]

Given that

\[ \int_{0}^{\frac{\pi}{2}} \sin^4 x \cos^2 x \, dx = \frac{\pi}{32} \]

we can use a property of definite integrals that states:

\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \]

### Step 1: Apply the Property

Let’s denote

\[ I = \int_{0}^{\frac{\pi}{2}} \cos^4 x \sin^2 x \, dx \]

Now, we will apply the property mentioned above:

\[ I = \int_{0}^{\frac{\pi}{2}} \cos^4 x \sin^2 x \, dx = \int_{0}^{\frac{\pi}{2}} \sin^4\left(\frac{\pi}{2} - x\right) \cos^2\left(\frac{\pi}{2} - x\right) \, dx \]

Using the identities \(\sin\left(\frac{\pi}{2} - x\right) = \cos x\) and \(\cos\left(\frac{\pi}{2} - x\right) = \sin x\), we can rewrite the integral:

\[ I = \int_{0}^{\frac{\pi}{2}} \sin^4 x \cos^2 x \, dx \]

### Step 2: Relate the Two Integrals

Now we have:

\[ I = \int_{0}^{\frac{\pi}{2}} \sin^4 x \cos^2 x \, dx \]

From the problem statement, we know:

\[ \int_{0}^{\frac{\pi}{2}} \sin^4 x \cos^2 x \, dx = \frac{\pi}{32} \]

### Step 3: Conclusion

Thus, we can conclude that:

\[ I = \frac{\pi}{32} \]

Therefore, the value of

\[ \int_{0}^{\frac{\pi}{2}} \cos^4 x \sin^2 x \, dx = \frac{\pi}{32} \]

### Final Answer

\[ \int_{0}^{\frac{\pi}{2}} \cos^4 x \sin^2 x \, dx = \frac{\pi}{32} \]

---